Consider a function $u(t, x, y)$ defined on $t \in (t_i, t_f) \ , x > 0 \ , y \in \mathbb{R}^{n - 2}$ satisfying the wave equation (here $\Delta_y$ is the Laplace operator w.r.t. the variable $y$) \begin{equation} \begin{aligned} -u_{tt}(t, x, y) + \partial_x^2u(t, x, y) + \Delta_y u(t, x, y) &= 0 \ , \end{aligned} \end{equation}
in the presence of two boundaries equipped with the boundary conditions \begin{equation} \begin{aligned} u(t_i, x, y) &= u_i(x, y) \ , \\ u(t_f, x, y) &= u_f(x, y) \ , \\ u(t, x, 0) &= f(t, x) \ , \\ \partial_x u(t, x, 0) &= g(t, x) \ . \end{aligned} \end{equation}
I wish to solve this PDE in terms of the functions $u_i \ , u_f \ , f$ and $g$.
Can anyone help me solve this? I'm confused due to the presence of two boundaries instead of one.